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| Description: Lemma 1 for frgrncvvdeq 30328. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.) | 
| Ref | Expression | 
|---|---|
| frgrncvvdeq.v1 | ⊢ 𝑉 = (Vtx‘𝐺) | 
| frgrncvvdeq.e | ⊢ 𝐸 = (Edg‘𝐺) | 
| frgrncvvdeq.nx | ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) | 
| frgrncvvdeq.ny | ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) | 
| frgrncvvdeq.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) | 
| frgrncvvdeq.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) | 
| frgrncvvdeq.ne | ⊢ (𝜑 → 𝑋 ≠ 𝑌) | 
| frgrncvvdeq.xy | ⊢ (𝜑 → 𝑌 ∉ 𝐷) | 
| frgrncvvdeq.f | ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) | 
| frgrncvvdeq.a | ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) | 
| Ref | Expression | 
|---|---|
| frgrncvvdeqlem1 | ⊢ (𝜑 → 𝑋 ∉ 𝑁) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | frgrncvvdeq.xy | . . . 4 ⊢ (𝜑 → 𝑌 ∉ 𝐷) | |
| 2 | df-nel 3047 | . . . . 5 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ 𝐷) | |
| 3 | frgrncvvdeq.nx | . . . . . 6 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) | |
| 4 | 3 | eleq2i 2833 | . . . . 5 ⊢ (𝑌 ∈ 𝐷 ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) | 
| 5 | 2, 4 | xchbinx 334 | . . . 4 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) | 
| 6 | 1, 5 | sylib 218 | . . 3 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) | 
| 7 | nbgrsym 29380 | . . 3 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) | |
| 8 | 6, 7 | sylnibr 329 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌)) | 
| 9 | frgrncvvdeq.ny | . . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) | |
| 10 | neleq2 3053 | . . . 4 ⊢ (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌))) | |
| 11 | 9, 10 | ax-mp 5 | . . 3 ⊢ (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌)) | 
| 12 | df-nel 3047 | . . 3 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌)) | |
| 13 | 11, 12 | bitri 275 | . 2 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌)) | 
| 14 | 8, 13 | sylibr 234 | 1 ⊢ (𝜑 → 𝑋 ∉ 𝑁) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 {cpr 4628 ↦ cmpt 5225 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Vtxcvtx 29013 Edgcedg 29064 NeighbVtx cnbgr 29349 FriendGraph cfrgr 30277 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-nbgr 29350 | 
| This theorem is referenced by: frgrncvvdeqlem7 30324 frgrncvvdeqlem8 30325 frgrncvvdeqlem9 30326 | 
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